# PROGRAMS AND ACTIVITIES

June 25, 2016

## September 9-11, 2011 Brock International Conference in Number Theory Brock University, St. Catharines, Ontario

 Invited Speaker Abstracts Contributed Talks Back to main index

Indices in a number field
by
(Université Calais, France)

Recents Progress In Dedekind Reciprocity Laws And L-Functions
by
(Paris, France)

In this talk we will discuss some recents results on the Dedekind reciprocity laws and their applications in numbers fields, finite fields and functions fields. Many interesting number-theoretical, algebraic, geometric, topological, differential, and combinatorial invariants are arising to theses laws will be studied.
New results will be given.

Computation of Galois groups via permutation group theory
by
Hugo Chapdelaine
(Université Laval)

In this talk we will present a method to compute the Galois group of certain polynomials defined over Q by combining the theory of local fields and the theory of permutation groups.

Recursions for weights of some Boolean functions in n variables
by
Thomas Cusick

University at Buffalo

We consider degree 3 Boolean functions in n variables which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. We provide an algorithm for finding a recursion for the truth table of any cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its (Hamming) weight as n increases. Given such a function, after the weights for the base cases are computed, we can find the weight for large n by a quick computation; it is not necessary to look at the truth table at all. Thus computations which previously seemed infeasible are now practical.

Stern polynomials and continued fractions
by
Karl Dilcher
, Dalhousie University
Coauthors: Kenneth B. Stolarsky

We derive identities for a polynomial analogue of the Stern sequence and define two subsequences of these polynomials. We obtain various properties for these two interrelated subsequences which have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. We also define two analytic functions as limits of these sequences. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points.

Finding the square roots of graphs
by
Brock University

Graph G is the square of graph H if two vertices u and v have an edge in G if and only if u and v are of distance at most two in H. Given H it is easy to compute its square H^2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H. In this talk we discuss the characterization and recognition problems of graphs that are squares of graphs of small girth.

Special values of Dedekind zeta-functions and motivic cohomology
by
Manfred Kolster
(McMaster University)

by
Claude Leverque
, U. Laval (Québec)

By definition, an integer n is said to be a congruent number if n is the area of a right angle triangle with rational sides: n=ab/2 where a^2+b^2 = c^2 with a, b, c being rational numbers. It turns out that the integer n is congruent if and only the elliptic curve E: Y^2 = X^3 - n^2 X has a rational solution (x,y) with xy non-zero. We will give other equivalent conditions under which n is a congruent number and we take this opportunity for exhibiting important properties of elliptic curves. This general audience lecture will be accessible to graduate and undergraduate students.

Minimal Zero Sum Sequences of Length Four over Finite Cyclic Groups
by
Yuanlin Li,
Brock University

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =\langle supp(S)\rangle$. The problem regarding the index of sequences has been studied in a series papers, and a main focus is to determine sequences of index 1. In this talk, we present the most recent development in this direction and show that if $G$ is a cyclic of prime power order such that $\gcd(|G|, 6)=1$, then every minimal zero-sum sequence of length 4 has index 1.

The index of an algebraic variety
by

Dino Lorenzini (Georgia University, USA)

Abstract: Let X be an algebraic variety X over a field K. The index of the variety is the greatest common divisor of the degrees of the points of the variety. When the variety is contained in the affine
space of dimension n, and (a_1,\dots, a_n) is a point with coordinates in the algebraic closure of K, the degree of this point over K is the degree of the field K(a_1,\dots,a_n) over K.
Thus a K-rational point has degree 1, and if a variety has a K-rational point, it has index one. The converse is false in general. We will survey some recent results on the index.

Norm index formula for the Tate Kernels
by

A. Chazad Movahhedi (Université Limoges, France)

Let F be a number eld and p an odd prime number. When F containsa primitive p-th root of unity p, the classical Tate kernel consists ofthe group of non-zero elements a 2 F such that fa; pg is trivial in
the second K-group K2F of the number eld F. We rst give a norm index formula for a "generalized" Tate kernel in a p-extension and then give some applications. This is a joint work with J. Assim.

The Fibonacci Zeta Function
by
Ram Murty
(Queens University)

The Fibonacci zeta function is the sum of the reciprocals of the k-th powers of the Fibonacci numbers. It is conjectured that all of these values are transcendental numbers, for k=1,2,... However, this is known only for even values of k. Suprisingly, the problem takes us into the world of modular forms, elliptic functions and modern transcendence theory. I will give a short survey of this fascinating chapter of number theory.

An example of modular form in three variables
by
Alexander Odesski
(Brock University)

We construct explicitely a modular form in three variables and discuss its properties.

Finding the square roots of graphs
by
Bill Ralph, Brock University

Is there an objective truth hidden within great works of art that only mathematics can explain? For hundreds of years we have revered and collected works of art by the great masters. I'll present statistical
evidence for a mathematical aesthetic shared by great artists across the centuries. I'll also show some of my own artwork and videos created using the mathematics of dynamical systems.

Rational approximation to real points on plane algebraic curves
by
Damien Roy
, University of Ottawa

Let C be a closed algebraic curve defined over Q in projective n-space. Assume that the set of real points of C with Q-linearly independent coordinates is infinite, and define lambda(C) to be the supremum of the uniform exponents of approximation of those points by rational points. Although Dirichlet's box principle simply shows that lambda(C) is at least 1/n, it is tempting to conjecture that it is always greater, i.e. that there always exist such points which are constantly much better approximated by rational points then expected from the box principle. At the moment, the only curves for which this is known to hold are the plane curves of degree 2. In this talk, we present heuristics which support the conjecture for all plane algebraic curves.

Well spaced integers generated by an infinite set of primes
by
Cam Stewart,
University of Waterloo

We shall discuss some joint work with Jeongsoo Kim which extends a result of Tijdeman from 1973 dealing with a question of Wintner. We prove that there exists an infinite set of prime numbers with the property that the
sequence of positive integers made up of primes from the set is well spaced.

On the size of the intersection of two Lucas sequences of distinct type
by
Alain Togbe
, Purdue University North Central
Coauthors: M. Cipu, M. Mignotte

The study of simultaneous Pell equations and generalizations has recently produced sharp bound for the number of solutions. In 2006, Bennett, Cipu, Mignotte, and Okazaki proved that for any fixed pair of distinct nonsquare positive integers $a$ and $b$ the system $x^2-ay^2=1, \; z^2-by^2=1$ has at most two solutions in positive integers. This is the best possible result one can have. In this talk, we will consider $a$ and $b$ two positive integers, with $a$ not perfect a square and $b>1$ and we will show that the Diophantine equation $x^2 - a(\frac{b^k-1}{b-1})^2 = 1$ has at most two solutions in positive integers. This extends a result obtained by He, Togb\'e, and Walsh. (The talk is based on a joint work with Cipu, Mignotte).

### Contributed Papers

The Prouhet-Tarry-Escott problem
by
Timothy Caley
, University of Waterloo

The Prouhet-Tarry-Escott (PTE) problem is a classical number theoretic problem which asks for integer solutions to sums of equal powers. Solutions to the PTE problem give improved bounds for the "Easier" Waring problem, but they are difficult to find using conventional methods. We will describe how solutions can be found computationally and by connecting the problem to finding rational points on elliptic curves. There will also be a statement of open questions relating to the PTE problem.

On a problem of Kowalski
by
Let $E$ be an elliptic curve defined over the rationals. For primes not dividing the conductor of $E$, we can reduce $E$ modulo $p$ to an elliptic cuve, denoted $E(F_{p})$. There exist integers $i(p)$ and
$f(p)$ such that $E(F_{p})$ is isomorphic to $\mathbb{Z}/i(p)\mathbb{Z} \times \mathbb{Z}/i(p)f(p)\mathbb{Z}$. Kowalski asked for the growth of $\sum_{p \le x} i(p).$